Twitux is a simple, clean and nice to look at (though still not too fancy) GNOME Twitter client.
Hides in the systray if closed, informs about new tweets using libnotify.
Although it is already very robust and stable (no crashes yet!), the user interface still has some bugs and annoyances, most of them around opening links found in Tweets.
But it's nevertheless already very usable and by far the best Twitter client I found for Linux.